Consider the rational map Ψ : [Special characters omitted.] where the fi's are homogeneous forms of the same degree in the homogeneous coordinate ring R = k[ x1,…,xd] of [Special characters omitted.]. Assume that I = (f 1,…,fm) is a height 2 perfect ideal in the polynomial ring R. In this context, the coordinate ring of the graph of Ψ is the Rees algebra of I and the co-ordinate ring of the image of Ψ is the special fiber ring. We study two settings. The first setting is when I is almost linearly presented. Here we study the ideal defining the graph and the image of Ψ. Whenever possible, we also study invariants such as the Castelnuovo-Mumford regularity and the relation type of the graph of Ψ. In the second setting we impose no constraints on the column degrees of the presentation matrix of I, but the number of generators of I is restricted to d + 1 (two more than the dimension of the source of Ψ). For this configuration, we study the image of Ψ.
We also introduce a new method, namely \"iterated'' Jacobian duals, to study the graph of Ψ. This is a generalization of the usual Jacobian duals which are often used to describe the graph of Ψ.
","abstract_html":"<p>Consider the rational map Ψ : [Special characters omitted.] where the <em>fi</em>'s are homogeneous forms of the same degree in the homogeneous coordinate ring <em>R</em> = <em>k</em>[<em> x</em>1,…,<em>xd</em>] of [Special characters omitted.]. Assume that <em>I</em> = (<em>f</em> 1,…,<em>fm</em>) is a height 2 perfect ideal in the polynomial ring <em>R.</em> In this context, the coordinate ring of the graph of Ψ is the Rees algebra of <em>I</em> and the co-ordinate ring of the image of Ψ is the special fiber ring. We study two settings. The first setting is when <em>I</em> is almost linearly presented. Here we study the ideal defining the graph and the image of Ψ. Whenever possible, we also study invariants such as the Castelnuovo-Mumford regularity and the relation type of the graph of Ψ. In the second setting we impose no constraints on the column degrees of the presentation matrix of <em>I</em>, but the number of generators of <em>I</em> is restricted to <em>d</em> + 1 (two more than the dimension of the source of Ψ). For this configuration, we study the image of Ψ.</p> <p>We also introduce a new method, namely "iterated'' Jacobian duals, to study the graph of Ψ. This is a generalization of the usual Jacobian duals which are often used to describe the graph of Ψ.</p>","abstract_has_math":false,"creators":["Mukundan, Vivek"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Bernd Ulrich","Giulio Caviglia","Edray H. Goins","William J. Heinzer"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-08-01T07:00:00Z","date_published":"2016-08-01T07:00:00Z","updated_at":"2026-07-24T03:54:02Z","subjects":["Pure sciences","Almost linearly presented ideals","Blowup algebras","Image of rational map","Iterated Jacobian duals","Rees algebras","Second analytic deviation one ideals","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://docs.lib.purdue.edu/open_access_dissertations/818","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bernd Ulrich","Giulio Caviglia","Edray H. Goins","William J. Heinzer"]},{"key":"dc:creator","label":"Author","values":["Mukundan, Vivek"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Pure sciences","Almost linearly presented ideals","Blowup algebras","Image of rational map","Iterated Jacobian duals","Rees algebras","Second analytic deviation one ideals","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://docs.lib.purdue.edu/open_access_dissertations/818"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Consider the rational map Ψ : [Special characters omitted.] where the fi's are homogeneous forms of the same degree in the homogeneous coordinate ring R = k[ x1,…,xd] of [Special characters omitted.]. Assume that I = (f 1,…,fm) is a height 2 perfect ideal in the polynomial ring R. In this context, the coordinate ring of the graph of Ψ is the Rees algebra of I and the co-ordinate ring of the image of Ψ is the special fiber ring. We study two settings. The first setting is when I is almost linearly presented. Here we study the ideal defining the graph and the image of Ψ. Whenever possible, we also study invariants such as the Castelnuovo-Mumford regularity and the relation type of the graph of Ψ. In the second setting we impose no constraints on the column degrees of the presentation matrix of I, but the number of generators of I is restricted to d + 1 (two more than the dimension of the source of Ψ). For this configuration, we study the image of Ψ.
We also introduce a new method, namely \"iterated'' Jacobian duals, to study the graph of Ψ. This is a generalization of the usual Jacobian duals which are often used to describe the graph of Ψ.
"]},{"key":"dc:title","label":"Title","values":["Rees algebras and iterated Jacobian duals"]}]}],"canonical_facts":{"dc:contributor":["Bernd Ulrich","Giulio Caviglia","Edray H. Goins","William J. Heinzer"],"dc:creator":["Mukundan, Vivek"],"dc:description.abstract":["Consider the rational map Ψ : [Special characters omitted.] where the fi's are homogeneous forms of the same degree in the homogeneous coordinate ring R = k[ x1,…,xd] of [Special characters omitted.]. Assume that I = (f 1,…,fm) is a height 2 perfect ideal in the polynomial ring R. In this context, the coordinate ring of the graph of Ψ is the Rees algebra of I and the co-ordinate ring of the image of Ψ is the special fiber ring. We study two settings. The first setting is when I is almost linearly presented. Here we study the ideal defining the graph and the image of Ψ. Whenever possible, we also study invariants such as the Castelnuovo-Mumford regularity and the relation type of the graph of Ψ. In the second setting we impose no constraints on the column degrees of the presentation matrix of I, but the number of generators of I is restricted to d + 1 (two more than the dimension of the source of Ψ). For this configuration, we study the image of Ψ.
We also introduce a new method, namely \"iterated'' Jacobian duals, to study the graph of Ψ. This is a generalization of the usual Jacobian duals which are often used to describe the graph of Ψ.
"],"dc:identifier":["https://docs.lib.purdue.edu/open_access_dissertations/818"],"dc:subject":["Pure sciences","Almost linearly presented ideals","Blowup algebras","Image of rational map","Iterated Jacobian duals","Rees algebras","Second analytic deviation one ideals","Mathematics"],"dc:title":["Rees algebras and iterated Jacobian duals"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:54:02Z"}